If you're still covered under TRICARE, then it probably covers counseling off-base, but it may not cover the specific therapist you want to see. You'd have to call and check.
As far as on-base, yes, you have to give your sponsor's info, but that doesn't mean the information is going to get back to him. In fact, you probably have a better expectation of privacy with on-base counseling because there may not be any insurance statement for your dad to read later (that is, unless on-base care for family members goes through TRICARE now too - never had any civilian dependents, so I wouldn't know.)
In general, though, if you want absolute confidentiality you have to pay for it yourself in cash.